Pruning to improve centre of tree

I have inherited a 20ft Magnolia grandiflora which has had little or no pruning over the years and which seems to have lost its central leader at some point. A lateral leader has taken over and as a result the tree centre is bereft of branches and foliage and has this strange branch bending out from the centre and growing out at an angle. It is very untidy indeed.

I would like to know whether I could remove this branch to encourage central growth and if so how far to the trunk I could take it? Would it produce new suckers or am I at risk of making the problem worse?

I have read that Magnolias can send out whippy growth that is non-flowering. How should I manage this new growth? I understand that pruning after the flowering period is the best time but this tree flowers throughout the summer - when do you recommend?